Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(5,𝔽2) | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,20)(9,23)(10,14)(11,17)(12,25)(13,24)(16,28)(18,27)(19,26)(21,29)(22,30)(31,34)(32,35)(33,36);(5,7,8,18,10,16,22,32)(6,11,12,27,14,26,29,35)(9,19,25,34,24,17,21,33)(13,15,20,31,23,28,30,36);(1,3,4,2)(7,16)(11,26)(15,28)(17,19)(18,32)(27,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,f^2=i,g^2=i,b^a=bd,f^a=fgi,g^a=gi,h^a=hi,c^b=ci,d^b=dg,g^b=gh,h^b=hi,f^d=fhi,g^d=gi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59104);